Tag Archives: Fractals: The Patterns of Chaos

We spent some more time investigating the Mandelbrot Set with with Fractal Zoomer, and have made a few observations:

The central “bulb” of the Mandelbrot Set is a cardioid, and each other bulb off of that central body is a perfect circle.

Each bulb has its own cycle of orbits. Some bulbs have the same size orbit as others, but with a different “order” (1,2,3,4,5,6 vs 1,3,5,2,4,6 for example)

Along the central “spike” on the left side of the Mandelbrot Set, we can find baby Mandelbrot Sets strung along that string. The main body of these baby Mandelbrots show cycles instead of fixed points. Realizing that the horizontal axis on which the Mandelbrot set is centered is the real number axis, this suggests that there may be a meaningful connection between the Mandelbrot Set and Feigenbaum Plot…

If you tile the plane with Julia Sets of a sufficient density, the collection of Julia Sets make a photo-mosaic of the Mandelbrot Set. This further reinforces that there is something important between the location of C within the Mandelbrot Set and the shape of the corresponding Julia Set.

This last point suggests there is a significant, meaningful connection between the location and shape of a Julia Set in the Mandelbrot Set. This will be what we’ll explore tomorrow.

For now, keep working with Fractal Zoomer and considering the questions posted yesterday. Please also read the section on Mandelbrot Set from pages 74-81 in your copy of Fractals: The Patterns of Chaos (and bring that book back on Thursday!).

We observed yesterday that cycles appear to be “born” in one of two ways: bifurcations of “lower” cycles, and spontaneously arising from chaos. We’ve already shown why cycles bifurcate, so we started today with an explanation about how cycles spontaneously emerge from chaos.

Referring to our previous proof, graphing the function y = f (f (x)) can be a way of finding new cycles. Fixed points on the graph of y = f (f (x)) that are not common with the graph of y = f (x) will be the parameters of our 2-cycle. By extension, graphing y = f (f (f (x))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us why cycles spontaneously appear from chaos. Explore the graph here. For a = 3.84, the “wiggliness” of the graph of y = f (f (f (x))) is enough for the fingers of the graph to touch the line y = x. But for a < 3.84, it isn’t. The moment that a becomes large enough for those fingers to touch the line y = x is the moment that a 3-cycle is born.

The graph above shows that it is not possible to get a 3-cycle before a = 3.84, meaning the 3-cycle “window” we see in the Feigenbaum Plot is the first time we get a 3-cycle (addressing one of the other questions we asked yesterday). It also gives a clue to the order of cycles. We’ve already noticed that the Feigenbaum Plot exhibits fractal-like self-similar behavior, and the 6-cycle we observed at a = 3.63 could almost be viewed as two groups of three. If we consider that the 3-cycle at a = 3.84 is “born” from the original fixed point trend we observed for a < 3.0, then we could argue that the 6-cycle is actually two conjoined 3-cycles, each born from the first bifurcation at a = 3.0. This would suggest that there is a 12-cycle for an even lower value of a, born from the second bifurcations 4-cycle (and indeed there is, at a = 3.5821).

The 5-cycle we see at a = 3.74 then is mirrored with a 10-cycle at a = 3.6053, and a 20-cycle at a = 3.5775. This pattern could continue forever, to find any cycle, of any length.

This argument forms the basis for the Sharkovskii order we saw in yesterday’s article. The 3 cycle is the very last cycle to be born out of the chaos of this trend. The 5-cycle is the second-to-last, and the 7-cycle and every other odd-numbered cycle comes before those. But before we get to any odd-numbered cycle, we first would find the 6-cycle (2 x 3). Before that, the 10 cycle (2 x 5); before that, the 14 -cycle (2 x 5), and so on. But before any of those, we find the 12-cycle (4 x 3); before that the 20-cycle (4 x 5); before that the 28-cycle (4 x 7). And so on, reading the Feigenbaum plot right-to-left, until we find our “un-bifurcating” powers of two cycles, stitching back together to 16, to 8, to 4, to 2, and then finally back to 1.

We discussed the reading from Fractals: The Patterns of Chaos and looked at some references to the “Butterfly Effect” in popular culture. We spent most of the rest of the period playing with the Solar System simulator online.

For Monday, please read pages 49-54 from your book, the section on “The Fractals and Chaos of Outer Space.”

We started watching a Nova documentary on The Strange New Science of Chaos. It’s from 1989, but it has held up well and serves as an excellent introduction to this strange new world of constrained randomness and sensitivity to initial conditions.

For Friday, there are three sections from Fractals: The Patterns of Chaos that I would like you to read:

Today, used the Box Count method to find again the dimension of Great Britain (report your findings here) then completed one last project to find calculate the dimension of one of the spiral fractal seen on the last dimension calculation sheet (this took most of the remainder of the period).

After discussing the reading from the text and the answer to yesterday’s question of the border between Spain and Portugal, we moved on to the last method of finding dimension, the Box Count method.

This method of finding dimension produces the same table of values and log-log plot that we made with the Richardson plot, but the values of S and C are found using a different method. Imagine overlaying a grid on top of a fractal image. We then count (C) the number of boxes of that grid that contain some portion of the fractal. We then repeat this process using a grid with smaller boxes, the sizes of which relative to the original give us S.

After enough counts are collected at different scales of boxes, we can create a log(c) vs log(s) plot and find the dimension using the slope as we did before. Your homework is to make the necessary counts with the coastline of Great Britain.